Line Shifts and Line Broadening in Light-Atom Interactions: Saturation, Rotating-Wave Approximation, and Spectral Shapes (MIT OCW)

MIT OpenCourseWare's lecture on light atom interaction presents how time dependent driving terms drive transitions between atomic states, and how absorption and stimulated emission arise from Fourier components at ±Omega. The instructor revisits the rotating wave approximation, explains the role of counter-rotating terms, and discusses how energy conservation emerges from the Schrödinger equation with a classical driving field. The talk connects the semi-classical field picture to a fully quantum description and introduces saturation, cross sections, and power broadening in a two level system. He then compares monochromatic and broadband excitation, explains Lorentzian line shapes, coherence time, and how line broadening and line shifts will be treated in the next chapter. The session includes practical links to experimental observables such as Bloch-Siegert shifts and absorption versus emission processes.

Overview

The lecture begins by establishing a time dependent Hamiltonian for light–atom interaction and shows that transitions between states occur when the driving term contains Fourier components at the right energy difference. Absorption corresponds to energy being supplied to the atom, while stimulated emission corresponds to energy being removed from the excited state, both governed by the phase factors in the evolving Schrödinger equation. The instructor emphasizes that energy conservation is encoded in the dynamics and that resonance occurs when the drive frequency matches the energy gap, with a finite time window allowing near resonance transitions within an uncertainty in energy and time.

Time-Dependent Driving and Absorption versus Emission

Using a semi-classical picture, the field couples to the atomic system through a dipole interaction. The integral over the interaction time shows that a nonzero transition amplitude requires a Fourier component of the drive matching the energy difference. The discussion connects this to the idea of photons entering or leaving the system and clarifies how the sign of the driving frequency selects absorption or emission processes, with the rotating frame playing a crucial role in simplifying the dynamics for near-resonant driving.

Rotating Wave Approximation and Counter-Rotating Terms

The talk revisits the rotating wave approximation (RWA) and explains when counter-rotating terms can be neglected. When the two-level system interacts with a circularly polarized field, angular momentum selection rules can suppress certain counter-rotating terms, effectively removing them from two-level physics in many common cases. However, if a higher-lying state is required by the selection rules, the counter-rotating term can become important. The lecturer connects this to the Bloch-Siegert shift and the broader energy-time uncertainty considerations, illustrating how the formalism naturally encompasses energy conservation and the role of the photon picture versus a semi-classical field.

Saturation and Power Broadening

The next major topic is saturation. The instructor defines the saturation parameter as the ratio of the driven excitation rate to the decay rate, showing how the excited-state population approaches a limit of one half under strong driving for a simple two-level system. In monochromatic driving, the unsaturated absorption rate scales with the square of the Rabi frequency, leading to a Lorentzian line shape whose height saturates and whose width broadens with power, a phenomenon known as power broadening. The discussion also covers broadband illumination, where the saturation intensity becomes independent of the transition strength because the spectral width of the light effectively samples the cross section and the atom’s spectral response.

Cross Sections and Monochromatic vs Broadband Light

Cross sections provide a geometric intuition for absorption as an effective shadow of the atom. For monochromatic light, the on-resonance cross section is set by the wavelength, while the saturation intensity depends on the transition frequency to the third power, reflecting the omega-cubed scaling of spontaneous emission. In broadband light, the cross section broadens with the transition, and the saturation intensity depends on the spectral shape of the light, not merely the intrinsic strength of the transition. The lecture then links these ideas to convolution of the atomic line shape with the light spectrum, illustrating how the saturation behavior can be similar in different regimes when the light’s spectral width is large enough.

Line Shapes, Coherence, and Time Scales

The speaker emphasizes that line shapes are determined by coherence times and by how long an atom can be interrogated coherently. In homogeneous broadening, a single mechanism (such as spontaneous emission or collisions) broadens all atoms in the same way, leading to an exponential decay of coherence and a Lorentzian line shape. In inhomogeneous broadening, a distribution of local environments or velocities yields a broadened ensemble line that may preserve narrow individual lines when observed with appropriate techniques. The correlation function formalism is introduced as a unifying language for these phenomena, with the time scale of the interrogation or the coherence time setting the observed width of the line. The discussion also touches on recoil, Doppler, and collisions as mechanisms that contribute to broadening in different contexts.

Towards Line Shifts and Broadening

The session closes with a roadmap to the next chapter on line shifts and line broadening, highlighting the interplay between external fields, dynamics of the driven system, and the spectral signatures that appear in spectroscopy. The instructor notes that many mechanisms can be understood in terms of finite observation time and coherence, and he prepares the audience to apply the correlation function formalism to quantify and interpret line shapes in experiments.